arXiv:1506.01074v3 [math.GR] 11 Mar 2016
Factoriality and the Pin-Reutenauer
procedure
J. Almeida
1∗J. C. Costa
2†M. Zeitoun
3‡1
Centro de Matemática e Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Portu-gal
2
Centro de Matemática e Departamento de Matemática e Aplicações, Universidade do Minho, Portugal
3
Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France
received 4thJune 2015, accepted 2ndFeb. 2016.
We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovarietyV of semigroups, the finitely generated free algebra for the
largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators.
Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.
Keywords: pseudovariety, profinite semigroup, profinite topology, topological closure, unary implicit signature, pure
implicit signature, rational language, aperiodic semigroup, Burnside pseudovariety, factorial pseudovariety, full pseu-dovariety, Pin-Reutenauer procedure
1
Introduction
Context and motivations. This paper deals with the computation of the closure of a given rational
language within a relatively free algebra, with respect to a suitable implicit signature and a profinite topology. A motivation for this line of research is the separation problem, which, given two rational languagesK and L, asks whether there is a rational language from a fixed class C containing K and disjoint fromL. The separation problem has several motivations. First, the membership problem for C reduces to the separation problem for C, since a language belongs to the class C if and only if it is separable from its complement by a language from C.
∗Work partially supported by CMUP (UID/MAT/ 00144/2013), which is funded by FCT (Portugal) with national (MCTES) and
European structural funds (FEDER), under the partnership agreement PT2020.
†Work supported, in part, by the European Regional Development Fund, through the program COMPETE, and by the Portuguese
Government through FCT, under the project PEst-OE/MAT/UI0013/2014.
‡Work carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the
frame of ANR 2010 BLAN 0202 01 FREC and of the "Investments for the future" Programme IdEx Bordeaux – CPU (ANR-10-IDEX-03-02).
Furthermore, solving this problem gives more information about the class under investigation, and is more robust when applying transformations to the class. For instance, is was proved by Steinberg (2001) and Place and Zeitoun (2015) that the classical operator V7→ V ∗ D on pseudovarieties preserves decid-ability of the separation problem, while it has been shown by Auinger (2010) that it does not preserve decidability of the membership problem (on the other hand, the status with respect to separation is un-known for other operators that do not preserve the decidability of membership, such as the power, as shown by Auinger and Steinberg (2003)).
Finally, deciding separation for some class can be used to decide membership for more involved classes: this is for instance a generic result in the quantifier alternation hierarchy, established by Place and Zeitoun (2014a), that deciding separation at levelΣnin this hierarchy entails a decision procedure for membership
at levelΣn+1.
Almeida (1999) has related the separation problem with a purely topological question, which is the main topic of this paper: the separation problem has a negative answer on an instanceK, L of rational languages if and only if the closures ofK and L in a suitable relatively free profinite semigroup, which depends on the class of separator languages we started from, have a nonempty intersection. Determining whether such closures intersect can be in turn reformulated in terms of computation of pointlike two-element sets in a given semigroup.
Deciding whether closures of rational languages intersect is often nontrivial, in particular because the profinite semigroup in question is uncountable in general. Yet, several classes of languages enjoy a prop-erty called reducibility(i)that states that the closures of two rational languages intersect in the suitable
relatively free profinite semigroup if and only if their traces in a more manageable universe also intersect. This more manageable universe may in particular be countable, and is therefore amenable to algorithmic treatment. In summary, reducibility is a property of the class of separators under investigation (or of the class of semigroups recognizing these separators), which reduces the search of a witness in the intersection into a simpler universe.
The most important example from the historical point of view is the class of languages recognized by finite groups. In this case, the relatively free algebra is the free group over some setX of generators, which is indisputably much better understood than the free profinite group overX. In particular, it is countable. Since it is known that the closures in the free profinite group of two rational languages intersect if and only if their traces in the free group also intersect (that is, the class of finite groups enjoys the reducibility property), this justifies the quest for an algorithm computing the closure in the free group of a rational language. Such an algorithm is known as the Pin-Reutenauer procedure, which we describe below, and has been developed along a successful line of research, see the work of Pin and Reutenauer (1991); Pin (1991); Ash (1991); Henckell et al. (1991); Ribes and Zalesski˘ı (1993); Herwig and Lascar (2000); Auinger (2004); Auinger and Steinberg (2005). As a consequence, the separation problem by group languages is decidable.
This framework can be generalized to classes consiting of other types of semigroups than just groups. Denote byκ the signature consisting of the binary multiplication and the unary (ω − 1)-power, with their usual interpretation in profinite semigroups. Note that the set of allκ-terms over X is isomorphic to the free group overX: the mapping sending each generator to itself and xω−1tox−1, the inverse ofx in the
free group, can be extended to a group isomorphism. More generally, given a pseudovariety V of finite semigroups, consider the semigroupΩκ
XV, which can be seen as the set of all interpretations over V of
κ-terms over X. This subalgebra of the pro-V semigroup over X is countable and thus, as said above, amenable to algorithmic treatment. One central problem in this context is theκ-word problem: given two κ-terms over X, decide whether they represent the same element in the relatively free profinite semigroup in the pseudovariety under consideration. This problem has already been investigated for several classical pseudovarieties besides that of finite groups, for instance by Almeida (1991, 1995) for the pseudovariety J of all finite J-trivial semigroups, by Almeida and Zeitoun (2004, 2007) for the pseudovariety R of all finite R-trivial semigroups, by Costa (2001) for the pseudovariety LSl of all finite semigroups whose local monoids are semilattices and by Moura (2011) for the pseudovariety DA of all finite semigroups whose regular J-classes are aperiodic semigroups. Moreover, reducibility has been shown to hold for several pseudovarieties, in particular by Almeida (2002) for J, by Almeida et al. (2005) for R, by Costa and Teixeira (2004) for LSl and, as already mentioned, by Ash (1991); Almeida and Steinberg (2000a) for the pseudovariety G of all finite groups. A further example is the pseudovariety A of aperiodic languages which, in a forthcoming paper, will be derived from the work of Henckell (1988), recently revisited by Place and Zeitoun (2014b, 2016), from which one can derive reducibility of this class. In other words, for these classes of languages, the separation problem reduces to testing that the intersection of the closures of two given rational languages in the suitable countable relatively free algebra is empty. This motivates designing algorithms to compute closures of rational languages in these relatively free algebras. This is one of the main contributions of this paper.
The Pin-Reutenauer procedure. In the core of the paper, we investigate how the profinite closure of
rational languages in free unary algebras interacts with concatenation and iteration. The natural guide for this work is provided by a procedure proposed by Pin and Reutenauer (1991) for the case of the free group. This procedure gives a way to compute a representation of the closure of a rational language inductively on the structure of the rational expression. Of course, the closure of a union is the union of the closures. The other two rules of the Pin-Reutenauer procedure deal with concatenation and iteration. For instance, when computing inΩκ
XV, the smallest subalgebra of the pro-V semigroup closed under multiplication and
(ω − 1)-power, establishing the Pin-Reutenauer procedure amounts to showing the following equalities: KL = K L,
L+= hLi κ,
whereL is the topological closure of L in Ωκ
XV, andhLiκis the subalgebra ofΩκXV generated byL.
No-tice that these equalities yield a recursive procedure to compute a finite algebraic representation ofL when L is rational. Such a finite representation may not immediately yield algorithms to decide membership inL for a given rational language L, but it reduces the problem of computing topological closures L to the problem of computing algebraic closureshLiκ. Since the signatureκ is finite, this representation also
provides a recursive enumeration of elements ofL. Additionally, assume that the following two properties hold:
(1) the word problem forκ-terms over V is decidable, (2) the pseudovariety V isκ-reducible.
Then one can decide the separation problem of two rational languagesK, L by a V-recognizable language. Indeed, Almeida (1999) has shown that this problem is equivalent to checking whether the closures ofK
andL in ΩXV intersect, which by reducibility is equivalent to checking whetherK ∩ L 6= ∅. In turn, this
may be tested by running two semi-algorithms in parallel:
(1) one that enumerates elements ofK and L and checks, using the solution over V of the word problem forκ-terms, whether there is some common element;
(2) another one that enumerates all potential V-recognizable separators.
Thus, the Pin-Reutenauer procedure is one of the ingredients to understand why a given class has decidable separation problem.
Contributions. It has been established recently by Almeida et al. (2014) that The Pin-Reutenauer
pro-cedure holds for a number of pseudovarieties. However, the results of this paper rely on independent, technically nontrivial results for the pseudovariety A of aperiodic semigroups: first, it was proved that the Pin-Reutenauer procedure is valid for A using the solution of the word problem for the free aperiodic κ-algebra given by McCammond (2001); Huschenbett and Kufleitner (2014); Almeida et al. (2015). Then, a transfer result was established to show that it is also valid for subpseudovarieties of A.
In this paper, we revisit the Pin-Reutenauer procedure, obtaining general results with simpler argu-ments. We consider unary signatures, made of multiplication and operations of arity 1. Our main result, Theorem 3.1, establishes that the Pin-Reutenauer procedure holds for the pseudovariety S of all finite semigroups, for unary signatures satisfying an additional technical condition, which is met forκ. The fact that rational languages are involved is crucial, since, as observed by Almeida et al. (2014, p. 10), the equalityKL = K · L fails for some languages K, L ⊆ X+, where closures are taken with respect to S
and the signatureκ.
This result is obtained by first investigating a property named factoriality. Factoriality of V with respect to, say, the signatureκ means that Ωκ
XV is closed under taking factors inΩXV. It was shown by Almeida
et al. (2014) that if V is factorial, then the Pin-Reutenauer procedure holds with respect to concatenation, that is,KL = K L for arbitrary K, L (not just for rational ones). However, it was also noted that the pseudovariety S cannot be factorial for nontrivial countable signatures, such asκ. In contrast, we show that any nontrivial pseudovariety of semigroups V closed under concatenation is factorial for the signature
1 consisting of multiplication and all unary operations. As an application, we obtain a new proof that the
minimum ideal of the free pro-V semigroup on at least two generators contains no 1-word. This property is a weaker version of a result obtained by Almeida and Volkov (2006). Besides the independent interest of such results, the technical tool used to prove them, named factorization history, is also the key to establish that the Reutenauer is valid for S. We further characterize pseudovarieties in which the Pin-Reutenauer procedure holds in terms of an abstract property named fullness, introduced by Almeida and Steinberg (2000a). The main idea is that the validity of the Pin-Reutenauer procedure for a pseudovariety V is inherited by a subpseudovariety W, as established by Almeida et al. (2014), provided both V and W are full. Conversely, we prove that if the Pin-Reutenauer procedure works for V, then V is full. Since the pseudovariety of all finite semigroups is full, this yields that a pseudovariety enjoys the Pin-Reutenauer property if and only if it is full.
Finally, we show that a variation of the Pin-Reutenauer procedure, known to hold in the case of all groups, also holds for pseudovarieties of groups in which every finitely generated subgroup of the free κ-algebra is closed.
Organization. The paper is organized as follows. In Section 2, we introduce the notion of history of
a factorization and we show that any nontrivial pseudovariety closed under concatenation product is 1-factorial. In Section 3, we establish that the Pin-Reutenauer property holds for S and unary signatures satisfying an additional condition. In Section 4, we relate the Pin-Reutenauer property with fullness, in the general case and in the case of pseudovarieties of groups.
2
1
-factoriality
We assume that the reader has some familiarity with profinite semigroups. For details, we refer the reader to the books of Almeida (1995); Rhodes and Steinberg (2009) and to the article of Almeida (2005). Here, we briefly introduce the required notation and key notions.
Preliminaries. Throughout the paper, we work with a finite alphabetX. For a pseudovariety V of semigroups, we denote byΩXV the free pro-V semigroup generated byX. Elements of ΩXV are called
X-ary implicit operations over V. See the paper of Almeida (1995) for details.
An implicit signature, as defined by Almeida and Steinberg (2000a), is a set of implicit operations of finite arity including the formal binary multiplication. Aσ-semigroup is an algebra in the signature σ whose multiplication is associative. Thus,σ-semigroups form a Birkhoff variety. We call an element of the freeσ-semigroup generated by X a σ-term. For convenience, we allow the empty σ-term.
Every pro-V semigroup has a natural structure ofσ-semigroup. We denote by Ωσ
XV the sub-σ-semigroup
ofΩXV generated byX. A σ-word over V is an element of ΩσXV. We denote by[_]Vthe surjective
homo-morphism ofσ-semigroups that associates to a σ-term t its interpretation [t]VinΩσXV. Whent is a word
and V is clear from the context, we writet instead of [t]V.
Unary implicit signatures. Let bN be the profinite completion of(N, +), i.e., the free profinite monoid on one generator. We denote by 1 the implicit signature consisting of multiplication together with all implicit operations _αwithα ∈ bN\ N. An implicit signature is called unary if it is contained in 1 and it
contains at least one unary implicit operation. For a unary implicit signatureσ, an element α ∈ bN such that theα-power operation _αbelongs toσ is said to be a σ-exponent. Note that by definition of 1, every
σ-exponent is infinite. An important example of a unary implicit signature is the signature κ, for which ω − 1 is the only κ-exponent.
Theσ-rank rankσ(t) of a σ-term t is the maximal nesting depth of elements of σ, disregarding
mul-tiplication, that occur int. It is defined inductively by rankσ(t1t2) = max(rankσ(t1), rankσ(t2)) and
rankσ(π(t1, . . . , tn)) = 1 + max1,...,n(rankσ(ti)) in case π is an operation from σ which is not
multipli-cation. For aσ-term
t = t0sα11t1· · · sαmmtm, (2.1)
where theti’s and thesj’s areσ-terms such that rankσ(ti) 6 rankσ(sj) = rankσ(t) − 1 and each αj is
aσ-exponent, we denote by νσ(t) the number m of subterms sαii oft. When σ is clear from the context,
we may writerank(t), ν(t) instead of rankσ(t), νσ(t), respectively.
Complete unary implicit signatures. A unary implicit signatureσ is said to be complete if the set of σ-exponents is stable under the mappingsα 7→ α − 1 and α 7→ α + 1. Note that 1 is complete, while κ is not. The intersection of a nonempty set of complete unary signatures either consists of multiplication solely, or
is again a complete unary signature. Therefore, the smallest complete unary signature containing a given unary signatureσ exists. It is called the completion of σ and it is denoted by ¯σ. By definition, we have σ ⊆ ¯σ, and a signature σ is complete if and only if ¯σ = σ. Note that for every 1-exponent α and every u ∈ ΩXS, the equalitiesuα−1= uαuω−1anduα+1= uαu hold. This proves the following useful fact.
Remark 1. Letσ be a unary signature containing κ. Then Ω¯σ
XV= ΩσXV.
2.1
Factorization sequences
Forα ∈ bN, we choose a sequence(ξn(α))n of natural integers converging toα. One can assume that
(ξn(α))n is constant if α is finite, or strictly increasing otherwise. Let t be a 1-term. We denote by
ξn(t) the word obtained by replacing each subterm vαwithα infinite by vξn(α), recursively. For instance,
ξn((aαb)β) = (aξn(α)b)ξn(β). The factorizationsξn(t) = x·y with x ∈ X∗andy ∈ X+may be obtained
recursively as follows:
• if rank(t) = 0, then ξn(t) = t for all n and there are |t| such factorizations of ξn(t);
• if rank(t) > 0 and t = t0sα11t1· · · smαmtm, where the ti’s and thesj’s are σ-terms such that
rank(ti) 6 rank(sj) = rank(t) − 1 (where the ti’s may be empty), then the factorizations ofξn(t)
are those of the following forms:
ξn(t) = ξn(t0sα11· · · tj−1sαjj) t′j· t′′jξn(s αj+1 j+1 tj+1· · · sαmmtm) (2.2) whereξn(tj) = t′jt′′j, and ξn(t) = ξn(t0sα11· · · s αj−1 j−1 tj−1skj)s ′ j· s ′′ jξn(sℓjtjsj+1αj+1· · · sαmmtm) (2.3) whereξn(sj) = s′js′′j,k, ℓ ∈ N, and k + ℓ + 1 = ξn(αj).
The conditiony ∈ X+, forbiddingy to be empty, is used recursively to ensure that each factorization of
ξn(t) is either of type (2.2) or (2.3), but not of both types: one can verify that each factorization of ξn(t)
is obtained by exactly one of the equations (2.2) and (2.3), wherej, t′
j, t′′j (in case (2.2)), orj, k, ℓ, s′j, s′′j
(in case (2.3)) are uniquely determined. In particular, the factorization
ξn(t0sα11· · · tp−1sαpptp) · ξn(sαp+1p+1tp+1· · · sαmmtm)
cannot be of type (2.2), since this would forcet′′
pto be empty, which is forbidden. This factorization is in
fact of type (2.3) withj = p + 1 and k = 0.
As an example, fort = aωbaω, the expression (2.1) is obtained form = 2, where t
0andt2are empty,
whilesα1
1 = aω,t1= b, and sα22= aω. Assumingξn(ω) = n!, we obtain
– the factorizationan!· ban!by (2.2), withj = 1, t′
1empty andt′′1 = b;
– the factorizationan!b · an!by (2.3),j = 2, k = 0, ℓ = n! − 1, s′
2empty ands′′2 = a.
The historyhn(t, x, y) of a factorization ξn(t) = xy is defined recursively as follows:
– ifrank(t) > 0 and the factorization is of the form (2.2), then hn(t, x, y) is obtained by
concatenat-ing the pair(1, j) with hn(tj, t′j, t′′j);
– ifrank(t) > 0 and the factorization is of the form (2.3), then hn(t, x, y) is obtained by
concatenat-ing the 4-tuple(2, j, k, ℓ) with hn(sj, s′j, s′′j).
The historyhn(t, x, y) is thus a word on an alphabet that depends on the integer n, which gives information
on how the wordξn(t) is split by the factorization xy. Note that the length of the history hn(t, x, y) is at
mostrank(t) + 1.
The simplified history shn(t, x, y) of the factorization ξn(t) = xy is obtained from the history hn(t, x, y)
by replacing each 4-tuple(2, j, k, ℓ) by (2, j). On the other hand, dropping the first two components of each letter of the historyhn(t, x, y), we obtain a word whose letters are pairs of nonnegative integers,
which we identify with an integer vector in even dimension, called the exponent vector. A factorization ofξn(t) can be recovered from its history but may not be recoverable from its simplified history without
the extra information contained in the exponent vector.
Observe that(ξn(t))nconverges to[t]SinΩXS. We will be interested in sequences(xn, yn)nsuch that
xnyn= ξn(t). We call such a sequence a factorization sequence for t.
It will be convenient in the proofs to work with factorization sequences having additional properties. Note that the set of simplified histories of factorizations ofξn(t) is finite and depends only on t. Moreover,
the dimension of all exponent vectors is bounded by2 rank(t). Therefore, any factorization sequence for t has a subsequence whose
(a) induced sequence of simplified histories is constant,
(b) induced sequence of exponent vectors belongs to Ndfor some constantd and converges in bNd\ Nd.
We call filtered a sequence with these properties. An application of this notion is the following simple statement.
Lemma 2.1. Let(xn, yn)nbe a factorization sequence for a 1-term. Then both (xn)n and(yn)nhave
subsequences converging inΩXS to 1-words.
Proof: Lett be the 1-term of the statement. By the above, one may assume that the sequence (xn, yn)n
is filtered. We proceed by induction onrank(t). The case rank(t) = 0 is straightforward. Otherwise, lett = t0sα1
1 t1· · · sαmmtmas in (2.1). There are two cases, according to the first letter of shn(t, x, y),
which can be of the form(1, j) or (2, j). Both cases are similar, so assume that it is of the form (1, j). Therefore, the factorizationxnynofξn(t) is given by (2.2), hence xn = ξn(t0sα11· · · tj−1s
αj
j ) t′j,nand
yn = t′′j,nξn(s αj+1
j+1 tj+1· · · sαmmtm) where ξn(tj) = t′j,nt′′j,n. By definition,tjis a 1-term and rank(tj) <
rank(t), whence by induction (t′
j,n)n and(t′′j,n)nhave subsequences converging to 1-terms, respectively
t′
jandt′′j. Therefore,(xn)n(resp.(yn)n) has a subsequence converging to the 1-term t0sα11· · · tj−1s αj j ·t′j (resp.t′′ j · s αj+1 j+1 tj+1· · · sαmmtm).
2.2
Factoriality of some pseudovarieties
A pseudovariety of semigroups is said to be closed under concatenation if the corresponding variety of rational languages has that property. A nontrivial pseudovariety V is closed under concatenation if and only if it contains A, the pseudovariety of aperiodic (or group-free) semigroups, and the multiplication of the profinite semigroupΩXV is an open mapping for every finite alphabetX as proved by Almeida and
Costa (2009) based on results of Straubing (1979) (in the monoid case) and Chaubard et al. (2006) (in the semigroup case) characterizing such pseudovarieties in terms of certain algebraic closure properties.
A pseudovariety V is saidσ-factorial if, for every finite alphabet X, every factor in ΩXV of aσ-word
over V is also aσ-word over V. Note that the pseudovariety S is not κ-factorial, since xαis a prefix ofxω
for everyα ∈ bN.
Theorem 2.2. Let V be a pseudovariety closed under concatenation. Then V is 1-factorial.
Proof: The statement is obvious if V is trivial. Otherwise, letu = vw be a factorization in ΩXV of an
arbitrary element ofΩ1
XV. Lett be a 1-term such that [t]V = u. Since the sequence (ξn(t))nconverges
tou = vw in ΩXV and the multiplication is open inΩXV, for all sufficiently largen, each ξn(t) may be
factorized asξn(t) = vnwnin such a way thatlim vn= v and lim wn= w.
By Lemma 2.1, both(vn)n and(wn)n have subsequences converging, in ΩXS, to 1-words over S.
Therefore, inΩXV, these subsequences converge to 1-words over V, so that v and w are actually 1-words
over V.
For a pseudovariety H of groups, H denotes the pseudovariety of all finite semigroups whose subgroups lie in H. In particular, when H is the trivial pseudovariety, then H= A. It is a well-known and elementary fact that H is always closed under concatenation. Denote by Bn the Burnside pseudovariety of all finite
groups of exponent dividingn. The pseudovariety Bn is thus defined by the pseudoidentityxω+n= xω.
In the following result, the special casen = 1, corresponding to the pseudovariety A, was first shown by Almeida et al. (2015) with a much more involved proof.
Corollary 2.3. For every positive integern the pseudovariety Bn isκ-factorial. In particular, the
pseu-dovariety A isκ-factorial.
Proof: We claim that the equalityΩκ
XBn = Ω1XBnholds for|X| = 1 and so also for every finite alphabet
X. Therefore, the result follows from Theorem 2.2. To prove the claim, we show that Ω{x}Bn = {x k |
k ∈ N} ∪ {xω, xω+1, . . . , xω+(n−1)}. For this, let α be a 1-exponent and let (a
k)k be a sequence of
integers converging toα. One can assume that ak modulon is a constant a, hence ak = nbk+ a with
bk ∈ N for all k. In Ω{x}Bn, we then havex
α = xω+α = lim
kxω+ak = limkxω+nbk+a = xω+a ∈
Ωκ XBn.
Another application of Theorem 2.2 is the following result, which is a weaker version of one that was established in (Almeida and Volkov, 2006, Corollary 8.12). Although the original result was formulated for the pseudovariety of all finite semigroups, the proof applies unchanged to pseudovarieties containing all finite local semilattices. Related results, under the same hypothesis as the following corollary, have been obtained by Steinberg (2010).
Corollary 2.4. If|X| > 2 and V is a nontrivial pseudovariety closed under concatenation, then there is no 1-word in the minimum ideal of ΩXV.
Proof: Since V is 1-factorial by Theorem 2.2 and since every element of ΩXV is a factor of every element
of the minimum ideal, if there were a 1-word in the minimum ideal then every element of ΩXV would be
a 1-word. We claim that this is impossible under the hypothesis that |X| > 2.
To prove the claim, observe that by definition, every 1-word of ΩXV which is not a word has at least
one infinite power of a finite word as an infix. In particular, it admits as factors powers of finite words of arbitrarily large exponent. Thus, it suffices to exhibit an element ofΩXV that fails this condition. For
this purpose letx, y ∈ X be distinct letters and consider the Prouhet-Thue-Morse substitution, defined byϕ(x) = xy, ϕ(y) = yx, and ϕ(z) = z for all z ∈ X \ {x, y}. This extends to a unique continu-ous endomorphism ofΩXV, which we also denoteϕ. Since, as proved by Hunter (1983), the monoid
of continuous endomorphisms ofΩXV is profinite under the pointwise convergence topology, we may
consider the elementϕω(x) = lim ϕn!(x). Now, it is well known that each word ϕn!(x) is cube free
(see, for instance, Lothaire (1983)). Since V is nontrivial and closed under concatenation product, it con-tains A. Therefore, the sets of the form(ΩXV)1u(ΩXV)1, whereu is a word, are open (Almeida, 1995,
Theorem 3.6.1). Hence,ϕω(x) is also free of cubes of finite words and so ϕω(x) is not a 1-word.
3
The Pin-Reutenauer procedure over
S
for pure signatures
Given a pseudovariety V of semigroups, an implicit signatureσ and a subset L ⊆ Ωσ
XV, we denote byL
the closure ofL in Ωσ
XV. Both the implicit signatureσ and the pseudovariety V are understood in this
notation. We are interested in computing a representation of such closures in two cases:
(a) when L is of the form pV(K) for some rational subset K of X+, wherepV is the natural continuous
homomorphism fromΩXS toΩXV;
(b) when L is a rational subset of Ωσ XV.
Recall that the class of rational subsets of a semigroup M is the smallest family of subsets of M containing the empty set and the singletons{m} for m ∈ M , and closed under union (Y, Z) 7→ Y + Z, product(Y, Z) 7→ Y Z = {yz | y ∈ Y, z ∈ Z} and iteration Y 7→ Y+ = S
k>1Yk. Since the
homomorphic image of a rational set is rational, any set of the form a is also of the form b. Conversely, there are of course rational sets ofΩσ
XV that are not obtained as image of a rational set ofX+underpV,
such as the singletons{aα} where α ∈ bN is aσ-exponent.
We say that the Pin-Reutenauer procedure holds for a class C of subsets ofΩσ
XV if, for everyK, L ∈ C,
the following conditions are satisfied:
KL = K L, (3.1)
L+= hLi
σ, (3.2)
wherehU iσdenotes theσ-subalgebra generated by the subset U of ΩσXV. Again, in this notation, the fact
that closures are taken inΩσ
XV is understood.
We say that V is (weakly)σ-PR if, for every finite alphabet X, the Pin-Reutenauer procedure holds for the class of all subsets ofΩσ
XV of the formpV(L) with L ⊆ X+a rational language. We say that V
is stronglyσ-PR if, for every finite alphabet X, the Pin-Reutenauer procedure holds for the class of all rational subsets ofΩσ
In this section, we only deal with the pseudovariety S. In Section 4, we shall transfer our results from S to other pseudovarieties. The main result of this section is the following theorem. It applies only to pure signatures, which we describe below.
Theorem 3.1. The pseudovariety S isσ-PR for every pure unary signature σ containing κ. The additional purity property thatσ is required to possess is the following.
Definition. A unary signatureσ is said to be pure if, for every positive integer d and for all α ∈ bN, ifdα is aσ-exponent, then α is also a ¯¯ σ-exponent.
Note that the quotient of dα by d is actually uniquely determined: if α, β ∈ bN and d ∈ N \ {0} are such that dα = dβ, then α = β. This follows immediately from the fact that the free profinite group on one generator, which is isomorphic to bN\ N, is torsion-free. Let us show this fact directly: dα = dβ means that all finite semigroups satisfy xdα = xdβ. To show that all finite semigroups also
satisfyxα= xβ, it is sufficient to consider 1-generated semigroups. Such semigroups have presentations
of the formSm,p = ha : am = am+pi, for integers m, p > 0. Note that the semigroup homomorphism
ϕ : Sm,p → Sdm,dpmappinga to adis injective. SinceSdm,dpsatisfiesxdα = xdβ, we have inSdm,dp
the equalitiesϕ(aα) = adα= adβ = ϕ(aβ), whence S
m,psatisfiesxα= xβ. This proves thatα = β.
In view of the following lemma, Theorem 3.1 can be applied to the signatureκ.
Lemma 3.2. The unary signatureκ is pure.
Proof: Everyκ-exponent is of the form ω + n, where n ∈ Z. Therefore, it suffices to show that, if n is an¯ integer,d is a positive integer, and α ∈ bN is such thatω + n = dα, then d divides n, whence α = ω +n
d is
again aκ-exponent. For that purpose, consider the unique continuous homomorphism of additive monoids¯ ϕ : bN→ Z/dZ which maps 1 to 1. We have ϕ(ω) = ϕ(limkk!) = 0 and ϕ(dα) = dϕ(α) = 0, and we
deduce from the equalityω + n = dα that ϕ(n) = 0.
To establish Theorem 3.1, we first prove a technical key lemma in Section 3.1. We shall then consider separately the cases of concatenation and iteration respectively in Sections 3.2 and 3.3.
3.1
A key lemma
We first prove a technical result which will be the key lemma in the sequel. It shows that, under suit-able hypotheses, one can balance the factors of a factorization of a givenσ-term to make them ¯¯ σ-terms themselves, without affecting membership in given clopen sets. ForL ⊆ X+, we denote bycl(L) the
topological closure ofL in ΩXS.
Given 1-terms t1, . . . , tm and languages L1, . . . , Lm, we say that (t1, . . . , tm) is a (L1, . . . , Lm
)-splitting of a 1-term t if the following conditions hold: (i) t = t1· · · tm;
(ii) [ti]S∈ cl(Li) for every i = 1, . . . , m.
Given aσ-term t, let λσ(t) = (rankσ(t), νσ(t)). We may write λ instead of λσwhenσ is clear from the
context.
Lemma 3.3. Letσ be a unary signature containing κ, let t be a ¯σ-term, and let L1, . . . , Lmbe rational
languages. Ift admits an (L1, . . . , Lm)-splitting (t1, . . . , tm), then there exists a ¯σ-term z admitting an
(1) [z]S= [t]S,
(2) ziis aσ-term for i = 1, . . . , m,¯
(3) λσ¯(zi) = λ1(ti) for i = 1, . . . , m.
Proof: We only prove the statement form = 2, since it is representative of the general case, and it allows a simplified notation.
Let(t1, t2) be an (L1, L2)-splitting of t. Set xn = ξn(t1) and yn = ξn(t2). By ii, lim xn = [t1]S
belongs tocl(L1), which is open by rationality of L1(Almeida, 1995, Thm. 3.6.1). Therefore, the word
xn belongs toL1for all sufficiently largen. Similarly, the word yn belongs toL2 for all sufficiently
largen. Let ϕ : X+ → S be a homomorphism into a finite semigroup recognizing both languages L1
andL2. In view of i,(xn, yn)nis a factorization sequence fort. Let (xnr, ynr)rbe a filtered subsequence
of(xn, yn)n, and let(k1,r, ℓ1,r, . . . , kd,r, ℓd,r)rbe the sequence of exponent vectors for the factorization
ξnr(t) = xnrynr. When(ki,r)r(resp.(ℓi,r)r) is constant, letki(resp.ℓi) be this constant value.
Other-wise, by taking a subsequence, we may assume that for eachs ∈ S, each of the sequences (ski,r)
rand
(sℓi,r)
ris constant, say with value respectivelyski andsℓi (i = 1, . . . , d), the integers ki andℓibeing
independent of the elements ∈ S.
In view of Case (2.3) of the definition of factorization sequence and sincet is a ¯σ-term, each sequence (ki,r+ ℓi,r+ 1)rconverges to someσ-exponent γ¯ i. In particular,γiis infinite. Define(αi, βi) by
(αi, βi) =
(ki, γi− ki− 1) if(ki,r)ris constant and(ℓi,r)runbounded,
(γi− ℓi− 1, ℓi) if(ki,r)ris unbounded and(ℓi,r)rconstant,
(γi− ℓi− 1, ω + ℓi) if both (ki,r)rand(ℓi,r)rare unbounded.
Note that in all cases, we have
αi+ βi+ 1 = γi. (3.3)
Letz1(resp.z2) be the 1-term obtained from xnr (resp. fromynr) by replacing for everyi the exponent
ki,rbyαiand the exponentℓi,rbyβi. Set
z = z1z2,
and let us verify thatz1, z2andz fulfill the desired properties. We have to show properties 1–3, and that
(z1, z2) is an (L1, L2)-splitting of z.
First note that, by (3.3), we have[yαi+βi+1]
S = [yγi]S for all 1-term y. Since (ki,r + ℓi,r + 1)r
converges toγi, using (2.3) we deduce that[z]S = [t]S, which proves 1. Next, by definitionαi andβi
either belong to N, or are of one of the formsγi− n or ω + n where n ∈ N. Since γiis aσ-exponent¯
and sinceσ contains κ, both αiandβiareσ-exponents, whence both z¯ 1,z2areσ-terms, which proves 2.¯
Finally, we haveλσ¯(zi) = λ1(ti) by construction, which is 3.
It remains to verify that(z1, z2) is an (L1, L2)-splitting of z. Condition i is satisfied by definition
ofz. Let us verify that z1 ∈ cl(L1) (showing that z2 ∈ cl(L2) is similar). Let ˆϕ : ΩXS → S be
the continuous extension of ϕ to ΩXS. By ii applied to the(L1, L2)-splitting (t1, t2) of t, we have
t1∈ cl(L1) = ˆϕ−1(ϕ(L1)). Since t1is the limit of(xnr)r, it suffices to show thatϕ(zˆ 1) = ϕ(xnr). This
follows from the claim that forr large enough, sξnr(αi)= ski = skir, which is clear if(k
ir)ris constant,
while it is obtained by reasoning in the group{sω+p| p > 0} if (k
3.2
The case of concatenation
We are now ready to treat the case of concatenation, that is, to establish Property (3.1).
Theorem 3.4. Equality (3.1) holds over the pseudovariety S for every unary signatureσ containing κ and for all rational languagesK, L ⊆ X+.
Proof: The inclusion from right to left in (3.1) amounts to continuity of multiplication inΩσ
XS and thus
it is always valid. For the direct inclusion, letv be an arbitrary element of KL. We need to show that v belongs toK · L.
Choose aσ-term t such that [t]S = v. Since v ∈ cl(KL) and since the closure cl(KL) of the rational
languageKL is clopen (Almeida, 1995, Thm. 3.6.1), the word ξn(t) belongs to KL for all sufficiently
largen. For such n, let t1,n∈ K and t2,n∈ L be words such that ξn(t) = t1,nt2,n, and let(t1,nr, t2,nr)r
be a filtered subsequence of(t1,n, t2,n)n. Fori = 1, 2, let tibe the term obtained by substituting each
exponent vector with the limit of the sequence of exponent vectors, in bNd, so thatlim ti,nr = [ti]S, and
(t1, t2) is a (K, L)-splitting of t. By Lemma 3.3, it follows that there exists a ¯σ-term z such that v = [z]S
andz admits a (K, L)-splitting (z1, z2) into ¯σ-terms. Since the unary signature σ contains κ, we have
Ω¯σ
XS= ΩσXS by Remark 1. Hence,[z1]S∈ cl(K) ∩ ΩXσ¯S= cl(K) ∩ ΩσXS= K, and similarly [z2]S∈ L.
Finally,v = [z]S= [z1z2]S= [z1]S· [z2]S∈ K · L.
3.3
The case of iteration
We now show that (3.2) holds over the pseudovariety S, for every pure implicit signatureσ containing κ and every rational languageL ⊆ X+. It is easy to see that the inclusion from right to left in (3.2) always
holds, see Almeida et al. (2014). The rest of this subsection is devoted to the proof of the other inclusion.
Theorem 3.5. Equality (3.2) holds over the pseudovariety S for every pure unary signatureσ containing κ and for every rational language L ⊆ X+.
The proof of Theorem 3.5 follows the lines of its analog for the pseudovariety A which is presented in (Almeida et al., 2014, Section 6), even though the argument requires significant changes in several points.
Consider an elementv of L+. We must show that there is a σ-term which coincides with v when
evaluated on (finitely many) suitable elements ofL. It turns out to be convenient to assume more generally thatv ∈ clσ¯(L+), so that there exists a ¯σ-term t such that [t]S= v. Therefore, we want to show that, for
everyσ-term t,¯
for every rational languageL, [t]S∈ L+implies[t]S∈ hLiσ. (Pt)
Letwk = ξk(t). The sequence of words (wk)kconverges tov = [t]S inΩXS. Asv belongs to the open
setL+, the wordw
k belongs toL+ for all sufficiently largek, and we may therefore assume that there
are factorizations
wk = w1,k· · · wrk,k, (3.4)
with eachwi,j∈ L. If there is a bounded subsequence of the sequence (rk)k, which counts the number of
factors fromL, then Theorem 3.4 yields that v belongs to the subsemigroup of Ωσ
XS generated byL and
we are done. We may therefore assume thatlim rk= ∞, which implies that rank(t) > 1. We first reduce
Proposition 3.6. Assume that (Pt) holds for everyσ-term t with ν(t) = 1. Then, it holds for every ¯¯ σ-term
t.
Proof: Lett be a ¯σ-term such that ν(t) > 1. Let v = [t]S, and assume thatv ∈ L+. To show thatv ∈ hLiσ,
we proceed by induction onλ(t), for the lexicographical order on N × N. Consider the factorization of t inσ-terms as in (2.1) and the factorization (3.4) of w¯ k= ξk(t). We distinguish three cases.
Case 1 Suppose first that there are infinitely many indicesk for which there exists ik ∈ {1, . . . , rk}
such that the first letter of the simplified history shk(t, w1,k· · · wi
k,k, wik+1,k· · · wrk,k)
is of one of the forms(1, j) with 0 6= j 6= m, or (2, j) with 1 6= j 6= m. That is, the corresponding factorizations(xk, yk), where xk = w1,k· · · wik,k andyk = wik+1,k· · · wrk,k, do not splitξk(t) in its
prefixξk(t0sα
1
1 ), nor in its suffix ξk(sαmmtm).
By Lemma 2.1, both (xk)k and(yk)k admit subsequences converging to 1-words, say v1 = [u1]S
andv2 = [u2]S respectively, with u1u2 = t. Since both xk and yk belong to L+, we deduce that
v1, v2∈ cl(L+). Therefore, one can apply Lemma 3.3: there exist ¯σ-terms z1andz2such thatv = [z1z2]S,
and fori = 1, 2, λ(zi) = λ(ui) and [zi]S ∈ cl(L+). By Remark 1, we obtain [zi]S ∈ L+. By the
assumption on the first letter of the simplified histories, we haverank(ui) = rank(t) and ν(ui) < ν(t),
henceλ(zi) = λ(ui) < λ(t) (i = 1, 2). Arguing inductively, we deduce that [z1]S and[z2]S belong to
hLiσ, whence so doesv = [z1z2]S= [z1]S· [z2]S. This concludes the proof for Case 1.
Case 2 Assume now that for all sufficiently largek, there is an index iksuch that the first letters of the
simplified histories
shk(t, w1,k· · · wi
k−1,k, wik,k· · · wrk,k)
shk(t, w1,k· · · wi
k,k, wik+1,k· · · wrk,k)
are of the forms (1,0) or (2,1) for the first one, and(1, m) or (2, m) for the second one. In other words, the factorwik,kofξk(t) jumps from the prefix ξk(t0sα11) to the suffix ξk(sαmmtm).
As in the first case, we may apply twice Lemma 2.1 to obtain from the following sequence of factoriza-tions
wk = w1,k· · · wik−1,k· wik,k· wik+1,k· · · wrk,k (k > 1),
an(L+, L, L+)-splitting of t into 1-terms, say t = u
1u2u3. Applying Lemma 3.3, we deduce that there
exists an(L+, L, L+)-splitting (z
1, z2, z3) into ¯σ-terms of a ¯σ-term z such that [z]S = v and λ(zi) =
λ(ui), i = 1, 2, 3. By Remark 1, we obtain [z1]S, [z3]S ∈ L+and[z2]S ∈ L. By the hypothesis of Case
2, we know that fori = 1, 3, we have either rank(ui) = rank(t) and ν(ui) = 1, or rank(ui) < rank(t).
Hence,λ(zi) < λ(t). Thus, we may apply the induction hypothesis to deduce that [z1]Sand[z3]Sbelong
tohLiσ. Hence, we finally havev = [z1]S· [z2]S· [z3]S∈ hLiσ· L · hLiσ⊆ hLiσ.
Case 3 Assume finally that for all sufficiently largek and for all indices ik, the first letter of the
simpli-fied history shk(t, w1,k· · · wik−1,k, wik,k· · · wrk,k) is
(a) either of the form (1,0) or (2,1), which means that wrk,kspans from the prefixξk(t0sα11) to the end
(b) or of the form (1, n) or (2, n), which means that w1,kjumps from the beginning ofξk(t) to the suffix
ξk(sαmmtm).
This case is treated as Case 2, settingv3(resp.v1) to be the empty term, in case a or b occurs.
To conclude the proof of Theorem 3.5, it remains to treat the case whereν(t) = 1. For dealing with this case, we use directed weighted multigraphs. A (multi)graph is a tuple(Q, (Ep,q)(p,q)∈Q×Q) where
Q is a set of vertices, and Ep,q is a set of edges having sourcep and target q, for each pair of vertices
(p, q) ∈ Q × Q. In the sequel, the graphs shall always be finite. A weighted multigraph is given by a multigraph together with a weight function, which associates to each edgee a nonnegative integer w(e). Ife is an edge with source p and target q, we represent this edge by
p−−−→ q.w(e)
For a path γ of a graph Γ, let c(γ) denote the edge-induced subgraph of Γ whose edges are those traversed byγ. We call c(γ) the support of γ. Furthermore, if ζ is an edge of Γ, then |γ|ζ denotes the
number of timesγ goes through the edge ζ. For a subgraph Γ′ofΓ, we denote by |γ|
Γ′ the minimum of
|γ|ζwithζ an arbitrary edge of Γ′.
Lemma 3.7. Let(πk)kbe a sequence of paths of a finite multigraphΓ. If there is some edge ζ for which
the sequence(|πk|ζ)kis unbounded, then there is some cycleγ such that (|πk|c(γ))kis unbounded.
Proof: Consider on each pathπk the subpaths which start with the edgeζ and whose length is the total
number of vertices of the graphΓ. Since there are only finitely many such subpaths, at least one of them, sayδ, must be used an unbounded number of times. Because δ must go at least twice through the same vertex,δ contains some cycle which satisfies the required condition.
We conclude the proof of Theorem 3.5 by establishing the following result, which, combined with Propo-sition 3.6, implies that Property (Pt) holds for every¯σ-term t.
Proposition 3.8. Property (Pt) holds for everyσ-term t with ν(t) = 1.¯
Proof: Lett be a ¯σ-term with ν(t) = 1. Let w = [t]S. Assuming thatw ∈ L+, we want to show that
w ∈ hLiσ. We havet = t0sα11t1, withrank(t0), rank(t1) 6 rank(s1) = rank(t) − 1. Let wk = ξk(t).
Since fork large enough, we have wk∈ L+, one may consider a factorization (3.4) ofwk. Using a similar
argument as in the proof of Case 2 of Proposition 3.6, we may assume, replacing(wk)kby a subsequence
if necessary, thatξk(t0) is a prefix of w1,kandξk(t1) is a suffix of wrk,k.
SinceL is a rational language of X+, there is a homomorphismϕ : X∗→ M onto a finite monoid M such thatϕ−1(1) = {1} and ϕ−1(ϕ(L)) = L. Let m and p be positive integers such that
am+p= amfor everya ∈ M . (3.5)
We construct for eachk a finite directed multigraph Γk. The set of vertices is
Vk=
(a, b) ∈ M × M : ξk(s1) ∈ ϕ−1(a)L∗ϕ−1(b)
∪ {ˆ, $},
where the two symbolsˆ and $ do not belong to M . The following are the edges of the graph Γk, where
• there is an edge (a1, b1) e+1
−−→ (a2, b2) if L ∩ ϕ−1(b1)ξk(se1)ϕ−1(a2) 6= ∅;
• there is an edge ˆ−→ (a, b) if L ∩ ξe k(t0se1)ϕ−1(a) 6= ∅;
• there is an edge (a, b)−−→ $ if L ∩ ϕe+1 −1(b)ξ
k(se1t1) 6= ∅.
Observe that, in view of (3.5), there is an edge in the graph of the formq1 −→ qe 2withe > m if and only
if there is also an edgeq1 e+p
−−→ q2ande + p 6 ξk(α1).
The purpose of this graph is to capture factorizations of the productξk(t0)ξk(s1)ξk(α1)ξk(t1) belonging
toL+. More precisely, for eachk, the factorizations
wk = ξk(t0)ξk(s1)ξk(α1)ξk(t1)
= w1,k· · · wrk,k
(3.6)
determine a path πk from vertex ˆ to vertex $: the factors wi,k which are not completely contained
in some factorξk(s1) determine the edges. Each intermediate vertex in the path corresponds to a factor
ξk(s1) together with a factorization into a word, followed by a possibly empty product of elements from L,
followed by a word, where only the values underϕ of the prefix and suffix words are relevant.
Conversely, every pathγ from ˆ to $ determines a factorization of a word of the form ξk(t0sℓ1t1) into a
product of elements ofL. Indeed, we may choose for each intermediate vertex q words uq,k, vq,k∈ X∗
andzq,k∈ L∗such that
ξk(s1) = uq,kzq,kvq,k. (3.7)
Then, for each edgeζ : qζ e+1 −−→ q′ ζ, the word yζ,k= vqζ,kξk(s e 1) uq′ ζ,k (3.8)
belongs toL. If the path γ is the sequence of edges (ζ0, ζ1, . . . , ζr), with ζ0: ˆ−→ qe ζ′0andζr: qζr
e+1
−−→ $, then we also have words
yζ0,k= ξk(t0s e 1) uq′ ζ0,k yζr,k= vqζr,kξk(s e 1t1)
inL. Then the following is the factorization associated with the path:
ξk(t0sℓ1t1) = yζ0,kzqζ1,kyζ1,k· · · zqζr,kyζr,k. (3.9)
The total numberℓ of factors ξk(s1) that are covered by following the path γ is the sum of the weights
of the edges, taking into account multiplicities; we call it the total weight of the path. Combining with Euler’s Theorem (Almeida, 1995, Theorem 5.7.1), it is now easy to deduce that each of the following transformations does not change the total weight of a path and therefore the value of the left side of the equality (3.9):
1. to traverse the edges in a path in a different order, without changing the number of times we go through each edge;
2. suppose that in the support of the path there are two cyclesδ1andδ2, with respective total weights
n1andn2, and that the positive integersr1andr2are such thatn1r1= n2r2; suppose further that
the path goes through each edge inδimore thanritimes; to replace the path by another one which
goes through each edge inδ1 lessr1times than before and through each edge inδ2morer2than
before;
3. if there are two edgesq1 e1
−→ q2andq3 e2+p
−−−→ q4in the path with bothe1, e2>m, to replace in the
path one occurrence of the edgeq1 e1
−→ q2by that of the edgeq1 e1+p
−−−→ q2, provided we compensate
by replacing one occurrence of the edgeq3 e2+p
−−−→ q4byq3 e2
−→ q4;
4. suppose that in the support of the path there is a cycleδ with total weight n and that the path goes through each edge in δ at least p + 1 times; suppose further that there is an edge q1
e
−
→ q2 with
weight at leastm; replace the path by another one which goes through each edge in δ less p times than before and change the edgeq1−→ qe 2byq1
e+np
−−−→ q2.
Using transformations of type 3, we may assume that the pathπkgoes through at most one edge whose
weight exceedsm + p − 1. Therefore, the remaining edges in πkare taken from a fixed finite set. Thus,
by taking a subsequence, we may further assume that all pathsπk use exactly the same edges of weight
at mostm + p − 1 and, either none of the πk use any other edges or, otherwise they all use only one
additional edgeζkconnecting two fixed vertices. Hence all the graphsc(πk) are the same finite graph, up
to an isomorphism that only changes one edge.
On the other hand, using transformations of type 4, we may assume that if all the pathsπkgo through
some edge of weight at leastm, then the graph c(πk) contains no cycle in which every edge is used at
leastp + 1 times.
We now split the argument into two cases. Suppose first that everyc(πk) contains an edge of weight at
leastm. In this case, one can apply Lemma 3.7 to deduce that there is a bound on the length of the paths πk and, therefore, we may assume that they all have the same length. Moreover, we may further assume
that, except for the edgeζi,k, at the same positioni, all paths πk = (ζ0, . . . , ζi−1, ζi,k, ζi+1, . . . , ζr) are
identical. Consider the factorizations
wk= yζ0,kzqζ1,k · · · yζi−1,kzqζi,k yζi,k,kzqζi+1,kyζi+1,k · · · zqζr,kyζr,k
of the form (3.9) associated with each of the pathsπk.
Letejbe the weight of each edgeζj(j 6= i) and let ei,kbe the weight of the edgeζi,k. Computing the
total weight, we obtain the formula
ξk(α1) = ei,k+
X
j6=i
ej. (3.10)
Lettingk → ∞ in (3.10), we deduce that lim ei,k = α1−Pj6=iej and, therefore,ei = lim ei,k is a
¯
σ-exponent, since so is α1and since by definition,σ is complete.¯
According to (3.8), the factorsyζj,kwithj /∈ {0, i, r} are given by
yζj,k= vqζj,kξk(s
ej
By compactness ofΩXS, we may assume that each of the sequences(uq,k)k,(vq,k)k, and(zq,k)k
con-verges to the respective limituq,vq, andzq(q = qζ1, . . . , qζr). Let
y0= [t0se10]Suqζ1 yr= vqζr[s er 1 t1]S yj= vqζj[s ej 1 ]Suqζj+1(j = 1, . . . , r − 1).
Then we obtain a factorization
w = y0zqζ1 · · · yi−1zqζi yizqζi+1yi+1 · · · zqζryr
in which eachyjbelongs tocl(L), while the zqbelong tocl(L+) ∪ {1}. By Lemma 3.3, we may assume
that eachuq,vq, andzq is aσ-word. By definition (3.7) of the words z¯ q,k, the latter has rank less than
rank(w). It follows that so is each yj. Hence theyj’s belong toL and the zqbelong toL+∪ {1}. By the
induction hypothesis, eachzqbelongs tohLiσ. Hence,w belongs to hLiσ, which completes the proof of
the first case.
It remains to consider the case where all edges have weight less thanm. By previous reductions, we know that the graphc(πk) is constant. Because the total weight tends to ∞, so does the length of the
pathπk. By Lemma 3.7, there is some simple cycleγ for which the sequence (|πk|c(γ))k is unbounded.
Applying transformations of type 2, we may assume that there is only one such cycle. By Lemma 3.7, we deduce that the pathsδ with c(δ) ⊆ c(πk) which go at most once through each edge in c(γ) have bounded
length. Hence, using transformations of type 1, we may assume that there is a pathδ from ˆ to $ such that the pathπk is obtained by inserting the power cycleγℓk at a fixed vertex in the pathδ, say πkis the
concatenated pathδ0γℓkδ1. Let the total weights of the pathsδiandγ be respectively niandn. Then the
total weight of the pathπkis given by the formula
ξk(α1) = n0+ n1+ nℓk. (3.11)
By taking a subsequence, we may assume that the sequence(ℓk)kconverges to someβ ∈ bN. From (3.11),
it follows thatnβ = α1− n0− n1. Since the signatureσ is assumed to be pure, we deduce that β is a
¯
σ-exponent.
By the argument in the preceding case, using the induction hypothesis, each of the pathsδi andγ
determines a corresponding element ofhLiσ, say respectivelyyiandy, such that w = y0yβy1. Sinceβ is
aσ-exponent, we may now end the proof by observing that it follows that w ∈ hLi¯ σ.
4
Pin-Reutenauer versus fullness
In this section, we apply the main results of Section 3 to show that the Pin-Reutenauer procedure is valid for many pseudovarieties. For this purpose, we establish relationships between that property and fullness, a notion introduced by Almeida and Steinberg (2000a). See also Almeida and Steinberg (2000b); Almeida et al. (2014) for related properties and other applications of fullness.
Recall thatpVdenotes the natural continuous homomorphism fromΩXS toΩXV. The pseudovariety
V is said to be full with respect to a class C of subsets ofΩσ
XS if the following equality holds for every
L ∈ C:
The closure of the left hand side of (4.1) is taken inΩσ
XS, while the closure of the right hand side is taken
inΩσ
XV. We say that V is (weakly)σ-full if, for every finite alphabet X, V is full with respect to the set of
all rational languages ofX+. We also say that V is stronglyσ-full if, for every finite alphabet X, V is full
with respect to the class of all rational subsets ofΩσ XS.
4.1
The general case
We first consider the case of arbitrary pseudovarieties and signatures.
Proposition 4.1. Letσ be an arbitrary implicit signature, V be a pseudovariety, and C be the closure under the rational operations of some set of finite subsets ofΩσ
XS. If the Pin-Reutenauer procedure holds
for C, then V is full with respect to C.
Proof: LetL be an arbitrary member of C. We need to show that the equality (4.1) holds. The inclusion from left to right is an immediate consequence of the continuity of the mappingpV. For the reverse
inclusion, we proceed by induction on the construction of a rational expression ofL in terms of finite sets in C. IfL is a finite set, then pV(L) = pV(L) and pV(L) = pV(L), and so the equality (4.1) is
trivially verified. Suppose next thatL1andL2are elements of C for which the equality (4.1) holds. Since
topological closure and the application of mappings commutes with union, the equality (4.1) also holds forL = L1∪ L2. On the other hand, we have the following equalities and inclusions:
pV(L1L2) = pV(L1) · pV(L2) since the Pin-Reutenauer procedure holds
for C
= pV(L1) · pV(L2) by the induction hypothesis
= pV(L1· L2) sincepVis a homomorphism
⊆ pV(L1L2) by continuity of multiplication.
Taking into account thatpVis a homomorphism ofσ-semigroups and that the inclusion from right to left
in (3.2) always holds (see the paragraph preceding Theorem 3.5, p. 12), one can similarly show that (4.1) holds forL = L+1.
The following is an immediate application of Proposition 4.1.
Corollary 4.2. Letσ be an arbitrary implicit signature and V a pseudovariety. If V is (strongly) σ-PR then V is (respectively strongly)σ-full.
The weak version of the following result is proved in (Almeida et al., 2014, Proposition 3.2). The same proof applies to the strong case.
Proposition 4.3. Let V and W be two (strongly) σ-full pseudovarieties such that V ⊆ W. If W is (respectively strongly)σ-PR, then so is V.
Note that S is triviallyσ-full for every implicit signature σ. Combining Theorem 3.1 with Corollary 4.2 and Proposition 4.3, we obtain the following result.
Corollary 4.4. Letσ be a pure unary signature containing κ. Then a pseudovariety V is σ-PR if and only if it isσ-full.
4.2
The group case
We now consider the case of pseudovarieties of groups, for the signatureκ.
Recall that a group is LERF (locally extendible residually finite) if every finitely generated subgroup is closed in the profinite topology. We say that a pseudovariety of groups H is LERF if, for every finite alphabetX, the relatively free group Ωκ
XH is LERF. By a classical result of Hall (1950), the pseudovariety
G of all finite groups is LERF.
By (Margolis et al., 2001, Proposition 2.9), G is in fact the only nontrivial extension-closed pseudova-riety of groups that is LERF. On the other hand, it is easy to check that, for the pseudovapseudova-riety Ab of all finite Abelian groups, every subgroup ofΩκ
XAb is closed (Delgado, 1998, Proposition 3.8).
A slightly different notion of stronglyκ-PR pseudovariety was considered by Pin and Reutenauer (1991) and Delgado (2001) (where it is simply called PR). Instead of property (3.2), the following property is considered:
L+= hLi
σ. (4.2)
Compared to (3.2), the topological closure in the right hand side of (4.2) has been dropped. As observed in (Almeida et al., 2014, end of Section 4), Equation (4.2) fails for the pseudovariety S and the implicit signatureκ, for L = a+b+, sinceaωb ∈ L+\ hLi
σ. However, the two notions coincide for the
pseudova-riety G (Pin and Reutenauer, 1991, Theorem 2.4). With same argument, we generalize this result to LERF pseudovarieties.
Lemma 4.5. Let V be a pseudovariety. If V satisfies (4.2) for a subsetL of Ωσ
XV, then (3.2) also holds.
If V is a LERF pseudovariety of groups andσ = κ, then (4.2) holds for rational subsets L of Ωκ XV.
Proof: Since forL ⊆ Ωσ
XV, the inclusionshLiσ ⊆ hLiσ ⊆ L+always hold, it is clear that (4.2) implies
(3.2).
For the second part, we argue as in (Almeida et al., 2014, p. 4). LetL be a rational subset of Ωκ XV.
Then,hLiκ= (L∪L−1)+is rational inΩκXV. By a well-known theorem of Anissimow and Seifert (1975),
the subgrouphLiκis therefore finitely generated. Hence,hLiκis closed inΩκXV, by the assumption that
V is a LERF pseudovariety. Finally, we haveL+ ⊆ hLi
κ, henceL+ ⊆ hLiκ, which, combined with the
reverse inclusion, which always holds, yields the result.
It can be shown easily that a κ-PR pseudovariety of groups is LERF (see Delgado (1997)). Thus, a pseudovariety of groups is stronglyκ-PR if and only if it is PR in the sense of Delgado (2001). In (Delgado, 2001, Corollary 3.9), it is also established that every “weakly PR” pseudovariety of groups is κ-full, a result which is considerably improved in the present paper, in the form of Corollaries 4.2 and 4.4.
It was conjectured by Pin and Reutenauer (1991) that G is stronglyκ-PR. Their conjecture was reduced to another conjecture, namely that the product of finitely many finitely generated subgroups of a free group is closed. The latter conjecture was established by Ribes and Zalesski˘ı (1993). Combining with Proposition 4.3 and Corollary 4.2, we obtain the following result.
Theorem 4.6. A pseudovariety of groups is stronglyκ-PR if and only if it is strongly κ-full.
The diagram in Fig. 1 summarizes the results of this subsection. We say that a pseudovariety of groups H is strong RZ if in every finitely generated free H-group, any finite product of finitely generated subgroups is closed. We say that H is weak RZ if, in every finitely generated free H-group, any finite product of finitely generated closed subgroups is also closed.
Strong κ-full Strong κ-PR Strong RZ LERF Weak κ-full Weak κ-PR Weak RZ Thm. 4.6 (Delgado, 1997, Thm. 4.2.1) Cor. 4.4
Fig. 1: Summary of results: pseudovarieties of groups
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